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Chapter 4 Exponential and Logarithmic Functions Copyright ©2015 Pearson Education, Ltd. All rights reserved. Section 4.1 Exponential Functions Copyright ©2015 Pearson Education, Ltd. All rights reserved. Would You Rather Have A Penny That Do...

Chapter 4 Exponential and Logarithmic Functions Copyright ©2015 Pearson Education, Ltd. All rights reserved. Section 4.1 Exponential Functions Copyright ©2015 Pearson Education, Ltd. All rights reserved. Would You Rather Have A Penny That Doubles Each Day For A Month Or $1 Million? Copyright ©2015 Pearson Education, Ltd. All rights reserved. Exponential Function  The exponential function is an important mathematical function which is of the form  f(x) = bx  Where b>0 and b1.  x is any real number. Copyright ©2015 Pearson Education, Ltd. All rights reserved. 𝒇 ( 𝒙 ) =𝟐 𝒙 If b > 1, then is an exponential growth function. x y (x,y) 3 8 (3,8) 2 4 (2,4) 1 2 (1,2) 0 1 (0,1) -1 (-1, ) -2 (-2, -3 (-3, Copyright ©2015 Pearson Education, Ltd. All rights reserved.  𝟏 𝒙 If 0 < b < 1, then is an 𝒇 ( 𝒙 ) =( ) exponential decay function. 𝟐 x y (x,y) 3 (3, 2 (2, 1 (1, ) 0 1 (0,1) -1 2 (-1,2) -2 4 (-2,4) -3 8 (-3,8) Copyright ©2015 Pearson Education, Ltd. All rights reserved. Copyright ©2015 Pearson Education, Ltd. All rights reserved.  Properties of Exponential Functions The domain is all real numbers ℛ The range is positive real numbers The graph is increasing (Growth) if b > 1 and decreasing (Decay) if 0 < b < 1 There is no x-intercepts, whereas there is only one y- intercept (0,1) The graph is continuous for all x. Copyright ©2015 Pearson Education, Ltd. All rights reserved. Exponential Rules Lows of Exponents Examples on Lows of Exponents Copyright ©2015 Pearson Education, Ltd. All rights reserved. Example: Solution: Copyright ©2015 Pearson Education, Ltd. All rights reserved. Copyright ©2015 Pearson Education, Ltd. All rights reserved.  To answer the question, we need to present the choices after calculation. The choices are: $1000 000 $5 368 709.12 Copyright ©2015 Pearson Education, Ltd. All rights reserved. Section 4.3 Logarithmic Functions Copyright ©2015 Pearson Education, Ltd. All rights reserved. Logarithmic Functions The logarithmic function is defined as For x > 0 , b > 0, and b ≠1, f(x) = logb x Copyright ©2015 Pearson Education, Ltd. All rights reserved. Logarithmic Functions The logarithmic function is the invers function to the exponential. Copyright ©2015 Pearson Education, Ltd. All rights reserved. Properties of Logarithmic Functions: The domain is positive real numbers The range is all real numbers ℛ The graph is increasing (Growth) if b > 1 and decreasing (Decay) if 0 < b < 1 There is no y-intercepts, whereas there is only one x- intercept (1,0). The graph is continuous for all x > 0. Example: Solution: Copyright ©2015 Pearson Education, Ltd. All rights reserved. Common Log and Natural Log Common Log: y = log(x) mean Natural Log: y = ln(x) mean y = Copyright ©2015 Pearson Education, Ltd. All rights reserved. Logarithmic Rules  To find log 10,000, ask yourself, “To what Example: exponent Since must 10 be raised to produce 10,000?” we see that 4 10 10000, Similarly, log 10,000 4. log1 0 because 100 1; 1 1 log.01  2 because 10 2  2  .01; 10 100 1 log 10  because 101/2  10. 2 Example: Find: log 100 Copyright ©2015 Pearson Education, Ltd. All rights reserved. Example: Solution: Section 4.4 Logarithmic and Exponential Equations Copyright ©2015 Pearson Education, Ltd. All rights reserved. Example: Find the value of x Solution:  Example: Find the value of Solution: Copyright ©2015 Pearson Education, Ltd. All rights reserved.  Solving an Exponential Equation Example: Solution:  Solving an Exponential Equation Example: Solution: Copyright ©2015 Pearson Education, Ltd. All rights reserved. Chapter 5 Mathematics of Finance Copyright ©2015 Pearson Education, Inc. All rights reserved. Section 5.1 Simple Interest and Discount Copyright ©2015 Pearson Education, Inc. All rights reserved. Simple Interest Copyright ©2015 Pearson Education, Inc. All rights reserved.  Simple Interest 3 months= year I = prt Since t is measured in years, 4 months= year t= 6 months= year 8 months= year 9 months= year Copyright ©2015 Pearson Education, Inc. All rights reserved.  To furnish her new apartment, Maggie Chan Exampl borrowed $4000 at 3% interest from her parents e: for 9 months. How much interest will she pay? Solutio Use the formula I Prt , 4000, r 0.03, and Pwith n: t 9 / 12 3 / 4 years : I Prt 3 I 4000 0.03  90. 4 Thus, Maggie pays a total of $90 in interest. Copyright ©2015 Pearson Education, Inc. All rights reserved. Simple Interest Copyright ©2015 Pearson Education, Inc. All rights reserved. Copyright ©2015 Pearson Education, Inc. All rights reserved. Simple Interest Copyright ©2015 Pearson Education, Inc. All rights reserved. Exampl John needs $9000 to pay for remodeling work on his e: house. A contractor agrees to do the work in 8 months. How much should John deposit now at 7% to accumulate the $9000 at that time? Solutio n: Copyright ©2015 Pearson Education, Inc. All rights reserved. Section 5.2 Compound Interest Copyright ©2015 Pearson Education, Inc. All rights reserved. Copyright ©2015 Pearson Education, Inc. All rights reserved.  Compound Interest 𝒏𝒕 𝒓 𝑨= 𝑷 (𝟏 + ) 𝒏 n=1 if the interest is compounded annually. n=2 if it is compounded semiannually. n=4 if it is compounded quarterly. n=12 if it is compounded monthly. n=365 if it is compounded daily. Copyright ©2015 Pearson Education, Inc. All rights reserved. Exampl If $1000 is invested at annual interest rate of 6%, e: compute the balance after 10 years if the interest rate is compounded: Semiannually, quarterly, and monthly. Solutio n: Copyright ©2015 Pearson Education, Inc. All rights reserved. Exampl If $1000 is invested at annual interest rate of 6%, e: compute the balance after 10 years if the interest rate is compounded: Semiannually, quarterly, and monthly. Solutio n: Copyright ©2015 Pearson Education, Inc. All rights reserved.  Compound Interest The compound amount A for a deposit of P dollars at an interest rate r per year compounded continuously for t years is given by: 𝑟𝑡 𝑃 = 𝑒 Copyright ©2015 Pearson Education, Inc. All rights reserved. Finding Doubling Time for an Investment Exampl e: Finding Time for an Investment to Double Exampl e: Solutio n: Finding Time for an Investment to Double To find the time for an investment to double: To find the time for an investment to triple: Copyright ©2015 Pearson Education, Inc. All rights reserved.  Chapter 6 Systems of Linear Equations and Matrices Copyright ©2015 Pearson Education, Inc. All rights reserved. Section 6.4 Basic Matrix Operations Copyright ©2015 Pearson Education, Inc. All rights reserved. What is a Matrix? The Size of a Matrix Types of Matrices Square Matrix Row Matrix Column Matrix Copyright ©2015 Pearson Education, Inc. All rights reserved. Examples of Matrices Applications Copyright ©2015 Pearson Education, Inc. All rights reserved. Examples of Matrices Applications Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: 3 2 1 1 ቂ ቃ = ൤ξ9 ξ4 3 ൨ 0 1 −2 0 1 ξ−8 Copyright ©2015 Pearson Education, Inc. All rights reserved. If A and B are both m × n matrices then the sum of A and B, denoted A + B, is a matrix obtained by adding corresponding entries of A and B. The difference of A and B, denoted A – B, is obtained by subtracting corresponding entries of A and B. Example: Complete the following operations on matrices  1  2 2  3 0 4  A   and B    0  1 3   2 1  4   2  2 6   4  2  2 A  B   A  B    2 0  1   2  2 7  Copyright ©2015 Pearson Education, Inc. All rights reserved. Exampl Find each sum if possible. e: (a) 5 A   6 and  4 B  6 8 9   8  3 Solution: Because the matrices are the same size, they can be added by finding the sum of each corresponding element.  5  6   4 6   5  (  4)  6  6  1 0 8 9    8  3  8  8 9  (  3)   16 6 .         (b) 5 8 3 9 1 A  and B  6 2  4 2 5 Solution: The matrices are of different sizes, so it is not A  B.to find the sum possible Copyright ©2015 Pearson Education, Inc. All rights reserved. If A is an m × n matrix and K is a scalar, then we let KA denote the matrix obtained by multiplying every element of A by k. This procedure is called scalar multiplication. Example: Complete the following operations on matrices if:  1  2 2  0  2 3 3 1  A   B   C    0  1 3  1 2 1  0  4 Copyright ©2015 Pearson Education, Inc. All rights reserved.   1  2 2  4  8 8  (a) 4 A 4      0  1 3   0  4 12   1   1 1 1 3 1  3  (b) C    3 3  0  4  0  4  3   1  2 2  0  2 3  3  2 0  (c) 3A  2 B 3    2     0  1 3  1 2 1   2  7 7  Copyright ©2015 Pearson Education, Inc. All rights reserved. Section 6.5 Matrix Products and Inverses Copyright ©2015 Pearson Education, Inc. All rights reserved. Multiplying two Matrices Copyright ©2015 Pearson Education, Inc. All rights reserved.  Multiplying two Matrices Find the product AB if  2 4 3  2 1  A   B   1 3   0 4  1   3 1   3(2)  2( 1)  1( 3) 3(4)  2(3)  1(1)  AB    0(2)  4(  1)  1(  3) 0(4)  4(3)  1(1)   5 7 AB     1 11 Classwork Copyright ©2015 Pearson Education, Inc. All rights reserved.   1 3   2 0 A   and B     2  7   3 4   1 3    2 0  7 12  AB         2  7  3 4    17  28    2 0  1 3   2  6  BA        3 4   2  7    5  19  AB ≠ BA Copyright ©2015 Pearson Education, Inc. All rights reserved. Matrix multiplication is not commutative Copyright ©2015 Pearson Education, Inc. All rights reserved.  Section 6.2 Larger Systems of Linear Equations Copyright ©2015 Pearson Education, Inc. All rights reserved. Write the Augmented Matrix of a System of Linear Equation Consider the following system of linear equations We can represent this system as The first column represents the coefficients of the variable , The second column represents the coefficients of the variable , the constants after the vertical line Copyright ©2015 Pearson Education, Inc. All rights reserved. Example { 2 𝑥+3 𝑦 − 6=0 4 𝑥 − 6 𝑦 + 2=0 { 𝑥 − 𝑦 + 𝑧 = 10 3 𝑥 +3 𝑦 =5 𝑥+2 𝑧 + 𝑦 =2  Write the System of Equations from the Augmented Matrix Example Solution Copyright ©2015 Pearson Education, Inc. All rights reserved. Copyright ©2015 Pearson Education, Inc. All rights reserved. Applying a Row Operation to an Augmented Matrix Example Solution Copyright ©2015 Pearson Education, Inc. All rights reserved. Copyright ©2015 Pearson Education, Inc. All rights reserved. The form of the augmented matrix in row echelon form is [ ] 1𝑎𝑏 𝑑 01 𝑐 𝑒 0 01 𝑓 Copyright ©2015 Pearson Education, Inc. All rights reserved. Copyright ©2015 Pearson Education, Inc. All rights reserved.  Example Solve the following system using matrices: { 𝑥+ 𝑦 + 𝑧 =30 2 𝑥+ 𝑧= 34 𝑦 + 𝑧 =16 Copyright ©2015 Pearson Education, Inc. All rights reserved. Copyright ©2015 Pearson Education, Inc. All rights reserved. Copyright ©2015 Pearson Education, Inc. All rights reserved. How to solve a system of linear equations using matrices Consistent Consistent Inconsistent Independent Dependent One solution Many solutions No solutions Q: Determine the type of each system, find the solution if there is any a) b) Example Solution  2 3 Evaluate:  2  1  4 3 2  12  10 4 1 Copyright ©2015 Pearson Education, Inc. All rights reserved. Copyright ©2015 Pearson Education, Inc. All rights reserved. Cramer’s Rule Copyright ©2015 Pearson Education, Inc. All rights reserved. Example Copyright ©2015 Pearson Education, Inc. All rights reserved. Solution Know Properties of Determinants Theorem The value of a determinant changes sign if any two rows (or any two columns) interchanged. Theorem If all the entries in any row (or any column) equal 0, the value of the determinant is 0. Theorem If any two rows (or any two columns) of a determinant have corresponding entries that are equal, the value of the determinant is 0. Theorem If any two rows (or any two columns) of a determinant is multiplied by a nonzero number k, the value of the determinant is also changed by a factor of k. Theorem If the entries of any row (or any column) of a determinant are multiplied by a nonzero number k and the result is added to the corresponding entries of another row (or column), the value of the determinant remains unchanged. Copyright ©2015 Pearson Education, Inc. All rights reserved. Example Copyright ©2015 Pearson Education, Inc. All rights reserved. Classwork Copyright ©2015 Pearson Education, Inc. All rights reserved.